Shortest descending paths through given faces

Ahmed, Mustaq; Lubiw, Anna

doi:10.1016/j.comgeo.2007.10.011

Cited by 10 publications

(16 citation statements)

References 8 publications

(4 reference statements)

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“…However, the similarities of an SDP with a shortest path discussed in Ahmed and Lubiw [3,4], particularly the uniqueness of an SDP through a given edge sequence [3,Lemma 7], gives us the belief that the bound holds:…”

Section: Structure Of An Sdp On a Convex Polyhedron Versus A Convex Tmentioning

confidence: 98%

“…The problem of computing a shortest descending path (SDP) was first addressed by Roy et al [8] who solved the problem for a convex terrain. Ahmed and Lubiw [3] gave an approximation algorithm for a terrain but in a restricted setting. Ahmed et al [1] devised two approximation algorithms for a general terrain, both based on the idea of discretizing the terrain by adding Steiner points.…”

Section: Introductionmentioning

confidence: 99%

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On the number of shortest descending paths on the surface of a convex terrain

Ahmed

Maheshwari

Nandy

et al. 2011

Journal of Discrete Algorithms

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The shortest paths on the surface of a convex polyhedron can be grouped into equivalence classes according to the sequences of edges, consisting of n-triangular faces, that they cross. Mount (1990) [7] proved that the total number of such equivalence classes is Θ(n 4 ). In this paper, we consider descending paths on the surface of a 3D terrain. A path in a terrain is called a descending path if the z-coordinate of a point p never increases, if we move p along the path from the source to the target. More precisely, a descending path from a point s to another point t is a path Π such that for every pair of points p = (x(p), y(p), z(p)) and q = (x(q), y(q), z(q)) on Π , if dist(s, p) < dist(s, q) then z(p) z(q). Here dist(s, p) denotes the distance of p from s along Π . We show that the number of equivalence classes of the shortest descending paths on the surface of a convex terrain is Θ(n 4 ). We also discuss the difficulty of finding the number of equivalence classes on a convex polyhedron.

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Section: Structure Of An Sdp On a Convex Polyhedron Versus A Convex Tmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

On the number of shortest descending paths on the surface of a convex terrain

Ahmed

Maheshwari

Nandy

et al. 2011

Journal of Discrete Algorithms

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“…We show how to compute the locally shortest path from p to q with respect to Σ under the modified metric by linear programming. Consider the case that every σ i is a triangle denoted by 3 . The case of some σ i being vertices or edges can be handled similarly.…”

Section: Locally Shortest Pathmentioning

confidence: 99%

“…For convenience, assume that v 0,j = p and v m+1,j = q for j ∈ [1,3]. We need the facet g of D * τi that contains the direction of the vector x i+1 −x i because the cost of x i x i+1 is equal to x i+1 − x i , n g / n g , n g , where ·, · denotes the inner product operator and n g denotes the vector that goes from the origin to a point in the support plane of g such that n g ⊥ g. By the convexity of D * τi , the facet f of D * τi that gives the largest x i+1 − x i , n f / n f , n f is the correct facet g. Therefore, we introduce a variable z i ∈ R and require…”

Section: Locally Shortest Pathmentioning

confidence: 99%

Navigating Weighted Regions with Scattered Skinny Tetrahedra

Cheng

Chiu

Jin

et al. 2017

Int. J. Comput. Geom. Appl.

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We propose an algorithm for finding a (1 + ε)-approximate shortest path through a weighted 3D simplicial complex T . The weights are integers from the range [1, W ] and the vertices have integral coordinates. Let N be the largest vertex coordinate magnitude, and let n be the number of tetrahedra in T . Let ρ be some arbitrary constant. Let κ be the size of the largest connected component of tetrahedra whose aspect ratios exceed ρ. There exists a constant C dependent on ρ but independent of T such that if κ ≤ 1 C log log n + O(1), the running time of our algorithm is polynomial in n, 1/ε and log(NW ). If κ = O(1), the running time reduces to O(nε −O(1) (log(NW )) O(1) ).

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“…Until recently the SDP problem has been studied in different restricted settings [2,13]. Two recent papers [3,14] give approximation algorithms for the problem on general terrains.…”

Section: Related Workmentioning

confidence: 99%

Shortest Gently Descending Paths

Ahmed

Lubiw

Maheshwari

2009

WALCOM: Algorithms and Computation

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Abstract. A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. We introduce a generalization of the shortest descending path problem, called the shortest gently descending path (SGDP) problem, where a path descends, but not too steeply. The additional constraint to disallow a very steep descent makes the paths more realistic in practice. We give two approximation algorithms (more precisely, FPTASs) to solve the SGDP problem on general terrains.

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Shortest descending paths through given faces

Cited by 10 publications

References 8 publications

On the number of shortest descending paths on the surface of a convex terrain

On the number of shortest descending paths on the surface of a convex terrain

Navigating Weighted Regions with Scattered Skinny Tetrahedra

Shortest Gently Descending Paths

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